Shop trigonometry and geometry

November 3, 2026  ·  Indumor

A ladder leaning against the shop wall is a right triangle with every letter in place: the floor is one leg, the wall the other and the ladder the hypotenuse. A 4 m ladder with its foot 1.5 m from the wall reaches 3.71 m up and stands at 68° to the floor. The same three sums turn a chamfer, a flange or the stock size for a hex into plain arithmetic, and this chapter puts them together with the tables that shorten them and three calculators.

The right triangle

The lettering stays fixed: a and b are the legs, c the hypotenuse, α the angle opposite leg a and β the angle opposite b. The two acute angles add up to 90°. Pythagoras ties the three sides together; sine, cosine and tangent tie an angle to two sides. Any two values give the other three, and the table says which formula does it in each case.

A check that catches most slips: sine and tangent grow with the angle, cosine shrinks. Tailstock offset and the compound angle for taper turning are this same triangle under other names, worked out in chapter 11.

Sines, cosines and tangents from 0° to 90°

The table runs in one-degree steps. For an angle with minutes, interpolate between two rows: the sine of 12° 30′ sits halfway between sin 12° (0.2079) and sin 13° (0.2250), that is 0.2164, which matches the exact value to four places. For sines and cosines, linear interpolation stays within a unit in the fourth decimal; for tangents the error grows with the angle, so above 60° use the calculator.

When the triangle has no right angle (a brace that does not meet the column square) the law of sines and the law of cosines take over, with sides a, b and c opposite angles α, β and γ:

Chord, sagitta and radius: what you measure for a rolling job

When a curved pipe or plate comes in to be copied, the radius cannot be measured directly: the center is in mid-air. What you can measure is the chord c, with a straightedge touching two points of the arc, and the sagitta h (the rise), the gap between the straightedge and the arc at the middle. Those two give the radius, the included angle and the developed length, which is what goes on the order to the plate roller.

Bolt circles

Flanges, covers and coupling plates carry their holes on a bolt circle of diameter D. With a DRO or a CNC you need the coordinates of each hole from the center: hole i sits at θ₀ + i · 360°/n, and the point comes from one cosine and one sine. For layout by hand you use the chord between neighboring holes, D times the constant k in the table, and step it around with dividers.

A detail that scraps parts: the first hole can sit at 0°, on the x axis, or at 90°, straight up. On a four-bolt flange that is the difference between holes on the cross and holes on the diagonal, and both are common. For six holes on D = 180 mm starting at 90°, the chord is 180 × 0.5 = 90 mm and the coordinates are (0, 90), (−77.942, 45), (−77.942, −45), (0, −90), (77.942, −45) and (77.942, 45). On the mill the same spacing comes from the dividing head (chapter 12).

Polygons: a hex or a square from round stock

The mill hand's question never changes: what round bar does a 24 mm across-flats hex need, or a 20 mm square? The corner-to-corner diagonal of a hexagon is 1.1547 times the across-flats size; the diagonal of a square is 1.4142 times the side. The bar has to be at least that diagonal, plus a little so the corners do not come out with unmilled flats: 27.7 mm for the 24 mm hex (start from 28 or 30 mm bar, or 1 1/8 in) and 28.3 mm for the 20 mm square (30 mm, or 1 1/4 in).

Any other regular polygon is in the general table, one factor per column. The interior angle is what the press brake needs: the faces of an octagon are bent to 180° − 135° = 45°.

Side = D × (column 4). Corner-to-corner diameter = across flats × (column 5). Area = side² × (column 6).

The sine bar

A sine bar is a ground bar with two identical rollers at an exact center distance, 5 or 10 in (100 or 200 mm in metric sets). With one roller on the surface plate and the other on a stack of gauge blocks of height h, the bar sits at an angle α such that sin α = h / L. To set an angle, build the stack to the height in the table; to measure one, adjust the blocks until a dial indicator reads the same along the part. Above 45° measure the complement instead, because the bar gets steep and the error grows.

Drawings give angles in degrees, minutes and seconds, and the table wants decimal degrees: 12° 30′ is 12.5°, and 7° 15′ 36″ is 7 + 15/60 + 36/3600 = 7.26°. Going back, 0.26° × 60 = 15.6′, and the fraction of a minute × 60 again = 36″. For 12° 30′ a 5 in sine bar needs 5 × 0.2164 = 1.0822 in of blocks; a 100 mm bar needs 21.644 mm.

Areas and volumes

The everyday formulas, for working out a part, a vessel or a flat blank before weighing it with the densities of chapter 22. Mind the units: an area in mm² divided by 100 is cm², and a volume in mm³ divided by 1000 is cm³.

The circular segment is the one that tells how much liquid is left in a horizontal tank at part load: radius and liquid depth give the wetted area and, times the tank length, the volume.

Frequently asked questions

How do you find the radius of an arc from the chord and the rise?

R = c² / (8·h) + h / 2. A 1200 mm chord with a 95 mm rise gives a radius of 1942 mm. The calculator in this chapter also returns the included angle and the arc length.

What size round stock do I need to mill a hexagon?

The corner-to-corner diagonal is 1.1547 times the across-flats size: a 24 mm hex needs 27.7 mm, so 28 or 30 mm bar; a 15/16 in hex needs 1.083 in, so 1 1/8 in bar. For a square, 1.4142 times the side.

How do I work out bolt circle coordinates?

With R = D/2 and each hole at θ = θ₀ + i·360°/n: x = R·cos θ, y = R·sin θ. For layout by hand, the chord between adjacent holes is D times the constant k in the table (0.5 for six holes).

Sources: ISO 80000-2:2019 (mathematical signs and symbols); ISO 80000-3:2019 (plane angle, radian and degree); Machinery's Handbook, Industrial Press (bolt circle coordinate tables, regular polygons and sine bar constants); elementary geometry (Pythagorean theorem, laws of sines and cosines). Every numerical table in this chapter is computed from those formulas.